On the Prevalence of Bridge Graphs Among Non-3-Connected Cubic Non-Hamiltonian Graphs

TL;DR

This paper investigates the prevalence of bridge graphs among non-3-connected cubic non-Hamiltonian graphs, proposing a conjecture and proving the claim for a specific class of graphs under that conjecture.

Contribution

It introduces a conjecture relating to bridge graphs and proves the claim for non-3-connected graphs assuming the conjecture holds.

Findings

Empirical evidence suggests most cubic non-Hamiltonian graphs are bridge graphs.

The paper proves the claim for non-3-connected graphs contingent on the conjecture.

Provides a new perspective on the structure of non-Hamiltonian cubic graphs.

Abstract

There is empirical evidence supporting the claim that almost all cubic non-Hamiltonian graphs are bridge graphs. In this paper, we pose a related conjecture and prove that the original claim holds for non-3-connected graphs if the conjecture is true.